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On the wave length of smooth periodic traveling waves of the Camassa–Holm equation()

This paper is concerned with the wave length λ of smooth periodic traveling wave solutions of the Camassa–Holm equation. The set of these solutions can be parametrized using the wave height a (or “peak-to-peak amplitude”). Our main result establishes monotonicity properties of the map [Formula: see...

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Detalles Bibliográficos
Autores principales: Geyer, A., Villadelprat, J.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Elsevier 2015
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4981178/
https://www.ncbi.nlm.nih.gov/pubmed/27546904
http://dx.doi.org/10.1016/j.jde.2015.03.027
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author Geyer, A.
Villadelprat, J.
author_facet Geyer, A.
Villadelprat, J.
author_sort Geyer, A.
collection PubMed
description This paper is concerned with the wave length λ of smooth periodic traveling wave solutions of the Camassa–Holm equation. The set of these solutions can be parametrized using the wave height a (or “peak-to-peak amplitude”). Our main result establishes monotonicity properties of the map [Formula: see text] , i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated with the equation, which distinguish between the two possible qualitative behaviors of [Formula: see text] , namely monotonicity and unimodality. The key point is to relate [Formula: see text] to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.
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spelling pubmed-49811782016-08-19 On the wave length of smooth periodic traveling waves of the Camassa–Holm equation() Geyer, A. Villadelprat, J. J Differ Equ Article This paper is concerned with the wave length λ of smooth periodic traveling wave solutions of the Camassa–Holm equation. The set of these solutions can be parametrized using the wave height a (or “peak-to-peak amplitude”). Our main result establishes monotonicity properties of the map [Formula: see text] , i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated with the equation, which distinguish between the two possible qualitative behaviors of [Formula: see text] , namely monotonicity and unimodality. The key point is to relate [Formula: see text] to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems. Elsevier 2015-09-15 /pmc/articles/PMC4981178/ /pubmed/27546904 http://dx.doi.org/10.1016/j.jde.2015.03.027 Text en © 2015 The Authors http://creativecommons.org/licenses/by/4.0/ This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Geyer, A.
Villadelprat, J.
On the wave length of smooth periodic traveling waves of the Camassa–Holm equation()
title On the wave length of smooth periodic traveling waves of the Camassa–Holm equation()
title_full On the wave length of smooth periodic traveling waves of the Camassa–Holm equation()
title_fullStr On the wave length of smooth periodic traveling waves of the Camassa–Holm equation()
title_full_unstemmed On the wave length of smooth periodic traveling waves of the Camassa–Holm equation()
title_short On the wave length of smooth periodic traveling waves of the Camassa–Holm equation()
title_sort on the wave length of smooth periodic traveling waves of the camassa–holm equation()
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4981178/
https://www.ncbi.nlm.nih.gov/pubmed/27546904
http://dx.doi.org/10.1016/j.jde.2015.03.027
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